Skip to main content
SpringerLink
Log in

Error Indicators for the Mortar Finite Element Discretization of a Parabolic Problem

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper we study residual spatial error indicators for a parabolic equation already discretized with respect to the time variable and approximated with the mortar finite element method. A posteriori error estimates are given at each step of time and are based on a local residual, the jumps of the normal derivative through the interfaces between elements and the jumps of the discrete solution through the mortars.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Babuška, R. Duran and R. Rodriguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular elements, SIAM J. Numer. Anal. 29 (1992) 947–964.

    Google Scholar 

  2. R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985) 283–301.

    Google Scholar 

  3. F. Ben Belgacem, The Mortar finite element method with Lagrange multipliers, Numer. Math. 84 (1999) 173–197.

    Google Scholar 

  4. F. Ben Belgacem, C. Bernardi, N. Chorfi and Y. Maday, Inf–sup conditions for the mortar spectral element discretization of the Stokes problem, Numer. Math. 85 (2000) 257–281.

    Google Scholar 

  5. A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of a nonlinear parabolic equation, submitted.

  6. C. Bernardi and F. Hecht, Error indicators for the mortar finite element discretization of the Laplace equation, Math. Comp. 71 (2002) 1371–1402.

    Google Scholar 

  7. C. Bernardi and Y. Maday, Mesh adaptivity in finite elements by the mortar method, Rev. Européenne Éléments Finis 9 (2000) 451–465.

    Google Scholar 

  8. C. Bernardi, Y. Maday and A.T. Patera, Domain decomposition by the mortar element method, in: Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, eds. H.G. Kaper and M. Garbey, NATO Advanced Science Institutes Series C, Vol. 384 (Kluwer Academic, Dordrecht, 1993) pp. 269–286.

    Google Scholar 

  9. C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: The mortar element method, in: Collége de France Seminar, Vol. XI, eds. H. Brezis and J.-L. Lions (Pitman, London, 1994) pp. 13–51.

    Google Scholar 

  10. C. Bernardi and B. Métivet, Indicateurs d'erreur pour l'équation de la chaleur, Rev. Européenne Éléments Finis 9 (2000) 425–438.

    Google Scholar 

  11. M. Bieterman and I. Babuška, The finite element method for parabolic equations I, A posteriori error estimation, Numer. Math. 40 (1982) 339–371.

    Google Scholar 

  12. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I, A linear model problem, SIAM J. Numer. Anal. 28 (1991) 43–77.

    Google Scholar 

  13. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems IV, Nonlinear problems, SIAM J. Numer. Anal. 32 (1995) 1729–1749.

    Google Scholar 

  14. F. Hecht and O. Pironneau, Multiple meshes and the implementation of Freefem+, Internal Report, I.N.R.I.A., Rocquencourt (1999).

    Google Scholar 

  15. R. Nochetto, G. Savaré and C. Verdi, A posteriori error estimates for variable time-step discretization of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000) 525–589.

    Google Scholar 

  16. R. Verfürth, A posteriori estimators for the Stokes equations, Numer. Math. 55 (1989) 309–325.

    Google Scholar 

  17. R. Verfürth, FEMFLOW-user guide, version 1, Internal Report, Universität Zürich (1989).

  18. R. Verfürth, A posteriori error estimator for nonlinear problem. Finite element discretizations of elliptic equations, Math. Comp. 62 (1994) 445–475.

    Google Scholar 

  19. R. Verfürth, A posteriori error estimates and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994) 67–83.

    Google Scholar 

  20. R. Verfürth, A posteriori error estimators and adaptive mesh-refinement for a mixed finite element discretization of the Navier–Stokes equations, in: Numerical Treatment of the Navier–Stokes Equations, eds. W. Hackbusch and R. Rannacher, Notes on Numerical Fluid Mechanics, Vol. 30 (Vieweg, Braunschweig, 1989) pp. 145–152.

    Google Scholar 

  21. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley and Teubner, 1996).

Download references

Author information

Authors and Affiliations

  1. Laboratoire SIANO, Département de Maths et d'Informatique, Faculté des sciences, Université Ibn Tofail, B.P. 133, Kénitra, Maroc

    A. Bergam & Z. Mghazli

  2. Laboratoire Jacques-Louis Lions, C.N.R.S. and Université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252, Paris Cedex 05, France

    C. Bernardi & F. Hecht

Authors
  1. A. Bergam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Bernardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Hecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Z. Mghazli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bergam, A., Bernardi, C., Hecht, F. et al. Error Indicators for the Mortar Finite Element Discretization of a Parabolic Problem. Numerical Algorithms 34, 187–201 (2003). https://doi.org/10.1023/B:NUMA.0000005362.95126.4c

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:NUMA.0000005362.95126.4c

Search

Navigation